The story is narrated in third person omniscient from Oliver's perspective.
Third person point of view is a type of narration that follows a character without that character directly telling the story. Instead of the main character as narrator telling the story with first person pronouns like “I” and “my,” third person narration has the main character described with his or her name and third person pronouns like “he” and “his.” An example of third person point of view follows.
Oliver stopped, and peeped into the garden. A child was weeding one of the little beds; as he stopped, he raised his pale face and disclosed the features of one of his former companions. Oliver felt glad to see him, before he went; for, though younger than himself, he had been his little friend and playmate. (Ch. 7)
In this quote, you can see that the main character is Oliver, because it is his actions we are following and his feelings we are told about. Notice that the narration uses either Oliver’s name or third person pronouns like “he” and “his.”
Third person contrasts with when the narrator, who is omniscient, addresses the reader directly. When this happens, the narrator uses first person pronouns like “I.”
Although I am not disposed to maintain that the being born in a workhouse, is in itself the most fortunate and enviable circumstance that can possibly befall a human being, I do mean to say that in this particular instance, it was the best thing for Oliver Twist that could by possibility have occurred. (Ch. 1)
You can see the difference in the pronouns. The narrator is describing Oliver, and speaking directly to the reader. Not all books have omniscient narrators that speak to the reader like this, but it is something that Charles Dickens liked to do, as if he were telling a story to the reader. It makes you close to the narrator as well as to Oliver. However it is Oliver’s story we are telling, not the narrator’s.
When an author chooses narration, he has to balance how close he wants the reader to be to his main character. Since Oliver is a child, we have an omniscient narrator, like the voice of Dickens himself, who is protective of Oliver. When we get Oliver’s perspective, it is always through the lens of this narrator with third person point of view. We are also treated to what other characters are thinking and doing, so that we can get Oliver’s full story.